Recently, attempts have been made to develop a robust method and system for predicting a function value from function values occurring earlier in time and second-derivative values of the function. Implementation of such a method and system can be illustrated as the prediction of current and future position values of an object based on previously determined position values along with current and past acceleration values. For example, S. O. Aase et al., “Compression Depth Estimation for CPR Quality Assessment Using DSP on Accelerometer Signals,” IEEE Trans. Biomed. Eng., vol. 49, pp. 263–268, March 2002., (hereinafter referred to as “Aase et al.”), incorporated herein by reference, describes the efforts of the authors to apply accelerometers to a mannequin in order to estimate chest movement using digital signal processing techniques on accelerometer signals. Aase et al. desired to provide a “practical, robust, and reliable solution for estimating chest compression depth.”
The proposed method of Aase et al. for estimating position from digital accelerometer data turned out to be unstable. The authors first estimated the digital velocity by using the trapezoidal rule, a classic Newton-Cotes polynomial-based numerical integration technique, to integrate accelerometer readings a(t) according to the following expression:
                              v          ⁡                      (            t            )                          =                                            ∫              0              t                        ⁢                                          a                ⁡                                  (                  τ                  )                                            ⁢                              ⅆ                τ                                              +                      v            ⁡                          (                              t                0                            )                                                          (        1        )            where t=nT and t0=(n−1)T, t is a time value, n is an integer, and T is the sampling interval. Applying the trapezoidal rule to perform the integration shown in (1) and converting the result from the analog time domain to the digital time domain produces an equation of the form:
      v    ⁡          [      n      ]        =                    T        2            ⁢              (                              a            ⁡                          [              n              ]                                +                      a            ⁡                          [                              n                -                1                            ]                                      )              +          v      ⁡              [                  n          -          1                ]            From the perspective of a DSP approach utilizing the z-transform, this operation can be viewed as a digital filtering operation employing a transfer function H(z) expressed as:
      H    ⁡          (      z      )        =            T      2        ⁢                  1        +                  z                      -            1                                      1        -                  z                      -            1                              Thus, the z-transform of the velocity function was found from the z-transform of the acceleration by multiplying the z-transform of the acceleration function by the transfer function H(z). Similarly, to arrive at the position function, the authors simply performed this operation a second time according to the equation X(z)=(H(z))2A(z). The position function obtained in this manner and expressed in the digital time domain is:
                              x          ⁡                      [            n            ]                          =                              2            ⁢                          x              ⁡                              [                                  n                  -                  1                                ]                                              -                      x            ⁡                          [                              n                -                2                            ]                                +                                                    (                                  T                  2                                )                            2                        ⁢                          (                                                a                  ⁡                                      [                    n                    ]                                                  +                                  2                  ⁢                                      a                    ⁡                                          [                                              n                        -                        1                                            ]                                                                      +                                  a                  ⁡                                      [                                          n                      -                      2                                        ]                                                              )                                                          (        2        )            
Although difference equation (2) could be implemented on a digital signal processor, it is based on an integration method (the trapezoidal rule) that was developed for polynomials, not oscillatory signals. Using the trapezoidal rule to integrate an acceleration signal results in position values that are independent of spectral information derived from an accelerometer signal. Neglecting the spectral information of an accelerometer signal results in a prediction equation such as that shown as expression (2), where the coefficient preceding each term is a constant. Expression (2), for example, has the constant coefficients:
  2  ,      -    1    ,      1    4    ,      1    2    ,            1      4        .  Using constant coefficients that are independent of the spectral information of the accelerometer signal causes the predicted function values to significantly deviate from the actual position values as shown in FIG. 1. This deviation is compounded for extended time periods as shown in FIG. 2.
The use of an error function to determine coefficients that reduce the divergence from the actual position explained above has been proposed in A. Sard, “Best Approximate Integration Formulas: Best Approximation Formulas,” Amer. J. Math., vol. 71, no. 1, pp. 80–91, 1949. Although the coefficients determined according to this publication may mitigate the amount of error in the calculated position values, these coefficients are also not based on the spectral content of a source signal.
Similarly, while the following two publications: R. B. Barrar et al., “Optimal Integration Formulas for Analytic Functions,” Bull. Amer. Math. Soc., vol. 79, pp. 1296–1298, 1973; and M. D. Stern, “Optimal Quadrature Formulae,” Comput. J., vol. 9, pp. 73–88, 1997, suggest methods including the minimization of an error integral, the methods are independent of spectral content, and they do not disclose the calculation of integration coefficients.
Accordingly, there is a need in the art for a robust method and system for determining current and future function values from past function values and past and current second-derivative values. The method and system should determine the current and future function values from at least a frequency of the second-derivative value of the function, and should minimize the deviation from the actual current and future function values over extended periods of time.